I am given to understand that the celebrated open problem (MLC) of the Mandelbrot set's local connectness has broader and deeper significance deeper than some mere curiosity of point-set topology.

From http://en.wikipedia.org/wiki/Mandelbrot_set I see that conjecture has implications concerning the structure of the Mandelbrot set itself, but I don't think I grasp its broadest implications for complex dynamics and matters beyond.

Request: Could someone explain the proper current context in which to view MLC and/or how the world would look it its full glory if MLC has a positive answer? Alternatively, please give a pointer to somewhere in the literature that does the same.

3 Answers
3

If a connected compact $K \subset C$ is locally connected then the Riemann map
$h\colon C \setminus \Delta \to C \setminus K$ extends continuously to $\partial \Delta$. For each $z \in K$, the boundary of the convex hull of $h^{-1}(\{z\})$ is the union of a set $\Lambda_z$ of chords; the union of these $\Lambda_z$ over all $z \in K$ is a closed set $\Lambda_K$ of disjoint chords; it is called a lamination of $\Delta$. We can reconstruct the convex hulls of each $h^{-1}(\{z\})$ from $\Lambda_K$, and when we collapse every convex hull to a point, we obtain a topological model for $K$.

In the case where $K$ is the Mandelbrot set $M$, the lamination $\Lambda_M$ can be described combinatorially, so MLC would mean that we know the topology of $M$.

There is a second answer which is more subtle and more important. For each $c \in M$, the filled Julia set $K_c$ of $z \mapsto z^2 + c$ is compact and connected; if it is locally connected the resulting lamination $\Lambda_c \equiv \Lambda_{K_c}$ is, in the right sense, invariant under $z \mapsto z^2$ on $\partial \Delta$. Even if $K_c$ is not locally connected, there is a way of defining what the lamination would be if $K_c$ were locally connected. Every invariant lamination appears as $\Lambda_c$ for some c, and MLC is equivalent to the statement there is a unique $c$ with a given lamination. We think of $\Lambda_c$ as describing the combinatorics of $K_c$, and we think of this uniqueness conjecture as "combinatorial rigidity"---two maps of the form $z \mapsto z^2 + c$ are conformally conjugate (and hence equal) if they are "combinatorially equivalent".

(Actually, if $z \mapsto z^2 + c$ has an attracting periodic cycle, then the set of combinatorial equivalent parameters form an open subset of $C$, so the statement of combinatorial rigidity must be suitably modified in that case. It is known that structural stability is open and dense in the family of maps $z \mapsto z^2 + c$, so combinatorial rigidity implies that every $z \mapsto z^2 + c$ in this open and dense set must have an attracting periodic cycle; this is the implication that Eremenko alluded to in his answer. )

In this sense MLC is closely analogous to Thurston's Ending Lamination Conjecture (proven by Brock, Canary, and Minsky), which says, broadly speaking, that a finitely generated Kleinian group is determined by the topology of its quotient and the ending laminations of its ends, which are also, when viewed appropriately, invariant laminations of the disk.

There is a third answer which is more historical and empirical. We can prove MLC and combinatorial rigidity "pointwise" (or "laminationwise") by proving that for a given invariant lamination $\Lambda$, it appears as the lamination $\Lambda_c$ for a single $c$. This has been done in great many cases, first by Jean-Christophe Yoccoz, and then by Mikhail Lyubich, the author of this post, Genadi Levin, and Mitsuhira Shishikura. To prove this combinatorial rigidity for a given $c$ seems to require a detailed understanding of the geometry of the associated dynamical system, and this almost always leads to further results. So proving MLC would most likely mean having a thorough understanding of the geometry and dynamics of every map $z \mapsto z^2 + c$.

I don't know enough to give a detailed answer, but I can at least give a reference: Milnor's book Dynamics in One Complex Variable (Vieweg). You can find an early draft here. Appendix F mentions a couple of conjectures that would follow from MLC: see the footnote on page F-2 in the linked pdf file.

While I'm at it, I can't resist repeating one of my favourite mathematical stories ever, which concerns the connectedness of the Mandelbrot set. It is connected, as you'd guess from the picture, but because of the thin filaments involved, this is not obvious if your image is too low-resolution. So there was some confusion in the early days. I'll let Milnor (Appendix F) take up the story:

Mandelbrot made quite good computer pictures, which seemed to show a number of isolated "islands". Therefore, he conjectured that [the Mandelbrot set] has many distinct connected components. (The editors of the journal thought that his islands were specks of dirt, and carefully removed them from the pictures.)

If you draw the Mandelbrot set by coloring a pixel black when the center of the pixel lies in the set, it will appear to be disconnected, and the islands will appear to be islands. This is true even if you draw it at a high resolution. It's only when you color in the pixels where the escape time is his that you see, very clearly, that the Mandelbrot set is connected.
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Jeremy KahnDec 28 '12 at 5:03

MLC is indeed a very technical and complicated counterpart of a simple question which arises from
the general theory of dynamical systems.
For a generic system (in a given finitely-parametric family) can one describe generic behavior of trajectories?

In our case "generic" means "open dense set". Our system is $z^2+c$ the simplest one parametric
family of one-dimensional rational (polynomial) maps.
For this case, the main question is: is it true, that there is an open dense set of parameters $c$,
such that for $c$ in this set, every trajectory that begins on an open dense set is converging
to an attractive cycle.

This is usually called the Density of Hyperbolicity Conjecture. For our family it is equivalent
to the MLC, but the equivalence is highly non-trivial.
Density of hyperbolicity is known if one restricts to real $c$.